Let be a group, a set, possibly infinite. Say has width with respect to if every element is the product of at most elements of .

Examples.

infinite, finite. Then width is infinite.

commutators , . Commutator width is 1 for finite simple groups.

Bi-invariant metrics. Let be the smallest subset of containing all generators and closed under conjugation. For instance, diffeomorphism groups, symplectomorphism groups have biinvariant metrics, and so infinite width with respect to . I have studied the case of Grigorchuk group.

2. Palindromes

Here, I will consider the set of palindromes.

Let be a free group. An element is a \textrm{palindrome} if it is reduced and reads the same in both directions.

Let be a finitely generated group, . An element of is a palindrom if one of its inverse images in is.

Example. . Then , are palindromes.

Notation. the reverse element of some presentation of .

Examples of groups with finite palindromic width.

Free metabelian groups (Bardakov-Gongopadhay, Riley-Sale).

certain solvable-by-nilpotent groups.

Examples of groups with infinite palindromic width.

Free groups (Bardakov-?).

Groups with free quotients.

3. Relations in groups

Proposition 1 Let be a non abelian group with no free quotient, generated by . Let be obtained from with one Nielsen transformation, and the adapted set of relators. Then there exists such that , in .

Proposition 2 The set is a normal subgroup of .

Theorem 3 (Fink-Thom) Let be a simple group generated by finite set . Then, with respect to , every element is a palindrome.

Indeed, the set is a non trivial normal subgroup, so .

Theorem 4 Let be a just infinite group generated by finite set . Then, with respect to , the palindromic width is finite.

4. Wreath products

A permutation wreath product is a semi-direct product .

Theorem 5 Let be a non abelian group. Assume that has palindromic width with respect to some subset . Then any element of is a product of at most palindromes with respect to generating set , where has generating set .